Bayesian ecology in R

• Scientific reasoning
  • How we reason
  • Motivated reasoning
• Bayes' theorem
  • Explanation
  • Terminology
  • Pros & Cons
• Prior knowledge
  
  • Example
• Frequentists vs Bayesians
  
  • Comparison of binomial tests
    
• Normal Linear model
  • Simulation
  • Inspection
  • Conclusions

Cancer case

Positive cancer test \( \neq \) cancer

# Variables
TP <- 0.9 #True Positive: 90%
FP <- 0.1 #False Positive: 10%

\[ P(cancer \mid positive test)= \frac {P(cancer )P(positive test \mid cancer))}{P(cancer )P(positive test \mid cancer))+P(no cancer)P(no cancer \mid positive test)} \]

prior <- 0.01 #Prior (prevalence of cancer in population): 1%

# Bayesian interpretation of the test
result <- (TP*prior)/(TP*prior+FP*(1-prior))

result*100

[1] 8.333333

• Bayesians are (more or less) ok with:

  Using prior knowledge
  Low sample size (rare species, lots of NAs, expensive sampling)
  Multiple comparisons (Geldman et al., 2012)
  Good way how to speed up science & research (significant results can be disproved)

• Upside

  More meaningful and comprehensive inferences (the only exact way how to draw inferences for generalized mixed models (Bolker et al., 2008)

• Downside

  Priors can disproportionally influence the posterior
  Choosing the right/appropriate prior can be challenging
  Computation heavy (less problem now)
  Garbage in = Garbage out

tests.freq[[1]]

    Exact binomial test

data:  c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 1, number of trials = 5, p-value = 0.4095
alternative hypothesis: true probability of success is greater than 0.1
95 percent confidence interval:
 0.01020622 1.00000000
sample estimates:
probability of success 
                   0.2 

tests.freq[[2]]

    Exact binomial test

data:  c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 2, number of trials = 10, p-value = 0.2639
alternative hypothesis: true probability of success is greater than 0.1
95 percent confidence interval:
 0.03677144 1.00000000
sample estimates:
probability of success 
                   0.2 

tests.freq[[3]]

    Exact binomial test

data:  c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 3, number of trials = 15, p-value = 0.1841
alternative hypothesis: true probability of success is greater than 0.1
95 percent confidence interval:
 0.05684687 1.00000000
sample estimates:
probability of success 
                   0.2 

tests.freq[[4]]

    Exact binomial test

data:  c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 4, number of trials = 20, p-value = 0.133
alternative hypothesis: true probability of success is greater than 0.1
95 percent confidence interval:
 0.07135388 1.00000000
sample estimates:
probability of success 
                   0.2 

• The true proportion of infected animals is 20 %.

data

 [1] 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
[36] 0 0 0 0 1

• We would need probably ALL Amur Leopards in the World to barely reject frequentist's H0 and do something to save them.
  

This is linked to statistical power of the test:

h<-pwr::ES.h(0.2,0.1) #true proportion is 20% and H0 proportion is 10%
pwr::pwr.p.test(h=h, power=0.8, sig.level=0.05, alternative = "greater")

     proportion power calculation for binomial distribution (arcsine transformation) 

              h = 0.2837941
              n = 76.76467
      sig.level = 0.05
          power = 0.8
    alternative = greater

• If the frequentist's H0 would be different, the outcome would be also different, which is not the case for Bayesian variant.
  

tests.bayes[[1]]

    Bayesian First Aid binomial test

data: c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 1, number of trials = 5
Estimated relative frequency of success:
  0.26 
95% credible interval:
  0.020 0.59 
The relative frequency of success is more than 0.1 by a probability of 0.886 
and less than 0.1 by a probability of 0.114 

tests.bayes[[2]]

    Bayesian First Aid binomial test

data: c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 2, number of trials = 10
Estimated relative frequency of success:
  0.24 
95% credible interval:
  0.039 0.48 
The relative frequency of success is more than 0.1 by a probability of 0.909 
and less than 0.1 by a probability of 0.091 

tests.bayes[[3]]

    Bayesian First Aid binomial test

data: c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 3, number of trials = 15
Estimated relative frequency of success:
  0.22 
95% credible interval:
  0.055 0.43 
The relative frequency of success is more than 0.1 by a probability of 0.931 
and less than 0.1 by a probability of 0.069 

tests.bayes[[4]]

    Bayesian First Aid binomial test

data: c(length(sub[[i]][sub[[i]] == "1"]), length(sub[[i]][sub[[i]] ==     "0"]))
number of successes = 4, number of trials = 20
Estimated relative frequency of success:
  0.22 
95% credible interval:
  0.071 0.40 
The relative frequency of success is more than 0.1 by a probability of 0.948 
and less than 0.1 by a probability of 0.052 

Bolker, B. M. (2008). Ecological models and data in R. Princeton University Press.

Bolker, B. M., Brooks, M. E., Clark, C. J., Geange, S. W., Poulsen, J. R., Stevens, M. H. H., & White, J. S. S. (2009). Generalized linear mixed models: a practical guide for ecology and evolution. Trends in ecology & evolution, 24(3), 127-135.

Gelman, A., Hill, J., & Yajima, M. (2012). Why we (usually) don't have to worry about multiple comparisons. Journal of Research on Educational Effectiveness, 5(2), 189-211.

Korner-Nievergelt, F., Roth, T., von Felten, S., Guélat, J., Almasi, B., & Korner-Nievergelt, P. (2015). Bayesian data analysis in ecology using linear models with R, BUGS, and Stan. Academic Press.